Let us call the probability, for a person of age x and in status j at time 0, of being in status k at time t. can be obtained by solving the ``forward Kolmogorov equations'', which decompose interval into and .
Introducing the 4 transition forces:
one can write for example:
These 6 differential equations can also be written (8):
Last equations, involving have been suppressed for simplicity. Two of the four equations are redundant, because . The system can therefore clearly be reduced to a set of two first order differential equations involving four forces and submatrix :